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arXiv:quant-ph/0102092

The quantum absolute phase observable

Gilad Gour

Published Feb 19, 2001• 0 citations
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Abstract

Defining the observable ${\bf ϕ}$ canonically conjugate to the number observable ${\bf N}$ has long been an open problem in quantum theory. Here we show how to define the absolute phase observable ${\bf Φ}\equiv |{\bfϕ}|$ by suitably restricting the Hilbert space of $x$ and $p$ like variables. This ${\bf Φ}$ is actually the absolute value of the phase and has the correct classical limit. A correction to the ``cosine'' ${\bf C}$ and ``sine'' ${\bf S}$ operators of Carruthers and Nieto is obtained.

DOI
10.48550/arxiv.quant-ph/0102092

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Defining the observable φ canonically conjugate to the number observable N has long been an open problem in quantum theory. The problem stems from the fact that N is bounded from below. Here we show how to define the absolute phase observable Φ ≡ |φ | by suitably restricting the Hilbert space of x and p like variables. This Φ is actually the absolute value of the phase and has the correct classical limit. A correction to the “cosine ” C and “sine ” S operators of Carruthers and Nieto is obtained.
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The well-known difficulties of defining a phase operator of an oscillator are considered from the point of view of the canonical transformation to action and phase-angle variables. This transformation turns out to be nonbijective, i.e., it is not a one-to-one onto mapping. In order to make possible the unitarity of its representations in quantum optics we should enlarge the Hilbert space of the problem. In this enlarged space we find a phase operator that, after projection, reproduces previous candidates to represent a well-behaved phase operator in the quantum domain.
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Based on the concept of classical phase,we formulate a new explanation for the quantum phase from the quantum mechanical point of view. The quantum phase is the canonically conjugate variable of an angular momentum operator,which corresponds to the angular position θ in an actual physical space with a classical reference frame,but it takes a complex exponential form eiθ ≡cosθ +i sinθ in the abstract Hilbert space of a quantum reference frame. This formulation is simply the famous Euler formula in a complex number field. In particular,when θ = π/2,the correlative quantum phase is a unitary pure imaginary number eiπ/2 ≡cos(π/2)+i sin(π/2) ≡ i. By using a photon state-vector function that is the general solution of photon Schrdinger equation and can completely describe a photon's behavior,we discuss the relationship between the angular momentum of a photon and the phase of the photon; we also analyze the intrinsic relationship between the macroscopic light wave phase and the microscopic photon phase.